Sounds like you may have a sticking solenoid. I wrote about this failure in my post Frigidaire Refrigerator Ice Dispenser Problems which gives a basic description of how the unit works and the potential failures. Give it a read and see if it helps.

Solenoid was bad (Plastic Melted). After replacing, still did not dispense Cubes. Voltage check shows a constant 50vac to solonoid even when not dispensing, probably why original burned out. Looking at print, should be dispenser board, but not sure. Anyone seen this one?

After replacing the expensive front control board, it still had the same problem. Fortunately it was in a rental home and the owner told the tenant to just live with crushed ice. I unplugged the wires to the solenoid to keep it from burning up too. The only part I didn't try was the control board in the back. If it fixes the problem for you, please let me know. I have a lot of $$$'s invested in this unit, with no return.